Dichotomy of entanglement depth for symmetric states

TL;DR

This paper reveals a dichotomy in the entanglement depth of symmetric states, showing they are either fully separable or fully entangled, and introduces a practical method to detect this in atomic ensembles, demonstrating high stability under noise.

Contribution

It establishes a dichotomy in entanglement depth for symmetric states and proposes an experimentally feasible detection method applicable to atomic systems.

Findings

Symmetric states are either fully separable or fully entangled.

The proposed detection method is stable under noise.

Dicke states can have large, stable entanglement depth.

Abstract

Entanglement depth characterizes the minimal number of particles in a system that are mutually entangled. For symmetric states, we show that there is a dichotomy for entanglement depth: an $N$-particle symmetric state is either fully separable, or fully entangled---the entanglement depth is either $1$ or $N$. This property is even stable under non-symmetric noise. We propose an experimentally accessible method to detect entanglement depth in atomic ensembles based on a bound on the particle number population of Dicke states, and demonstrate that the entanglement depth of some Dicke states, for example the twin Fock state, is very stable even under a large arbitrary noise. Our observation can be applied to atomic Bose-Einstein condensates to infer that these systems can be highly entangled with the entanglement depth that is of the order of the system size (i.e. several thousands of atoms).